Heterogenous Competition in a Differentiated Duopoly with Behavioral Uncertainty

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1 Heterogenous Competition in a Differentiated Duopoly with Behavioral Uncertainty Akio Matsumoto Department of Systems Industrial Engineering Univesity of Arizona, USA Department of Economics Chuo University, JPN Ferenc Szidarovszky Department of Systems Industrial Engineering University of Arizona, USA Abstract This study examines how uncertainty about rival firm s behavior affects the equilibrium its dynamics under heterogenous competition A differentiated mixed duopoly is considered in which one firm sets a quantity the other firm charges a price It is demonstrated that behavioral uncertainty stabilizes the unstable dynamic process in comparison to behavioral certainty generates Pareto-improvement in the sense that the firms can earn larger profits Keywords: heterogenous strategy, asymmetric product differentiation, behavioral uncertainty, mixed duopoly The paper was prepared when the first author visited the Department of Systems Industrial Engineering of the University of Arizona He appreciated its hospitaliby over his stay The usual disclamier applies Tucson, AZ , USA, akiom@ arizonaedu, 74-, Higashinakano, Hachioji, Tokyo, , JPN, akiom@tamaccchuo-uacjp Tucson, AZ , USA, szidar@siearizonaedu

2 Introduction We study a mixed duopoly model with incomplete or partial information in which one firm adjusts its quantity the other firm its price but each firm lacks knowledge of the rival firm s strategic behavior (ie, quantity or price strategy) The mixed duopoly is an intermediate model between a pure Cournot duopoly a pure Bertr model, which was first considered by Bylka Komar (976) then extended in various directions Using a symmetric differentiated duopoly proposed by Dixit (979), Singh Vives (984) showed, among others, that the quantity-adjusting strategy is more (less) profitable than the price-adjusting strategy if the goods are substitutes (complements) Recently Matsumoto Szidarovszky (008) constructed a n-firm differentiated mixed oligopoly model with n> showing that their result is sensitive to the duopoly assumption Szidarovszky Molnár (99) proved the equivalence of the equilibrium to the non-linear complementarity problem also proved the existence uniqueness of the Nash equilibrium in a general N-firm model These, only to name a few, are static studies Matsumoto Onozaki (005) Yousefi Szidarovszky (005) modeled the dynamic process of mixed duopolies with nonlinear dems showed the birth of complicated fluctuations In this paper we return to the duopoly model assume uncertainty about the type of strategic variable chosen by the rival firm We call a situation in which each firm has complete information about the behavior of the rival firm the market dem fully-informed the one in which each firm has incomplete information about the behavior of the rival firms but complete information about the market dem partially-informed In the recent literature it has been demonstrated that various phenomena may occur as a result of the lack of information about the market dem Bischi et al (004) construct a Cournot duopoly in which Nash equilibrium is unique under perfect information find the emergence of self-confirming steady state present a situation in which global bifurcation of basin of attraction occurs when firms choose their actions based on a misspecified market dem Bischi et al (007) examine a quantity adjustment process of a nonlinear duopoly model show its stability under incomplete information about the market dem, despite the fact that it can generate complicated dynamics under complete information when the nonlinearities become stronger In the existing literature, however, very little has been done with respect to the effects caused by the behavioral uncertainty on the equilibrium of the mixed duopoly its dynamic process The main purpose of this study is to shed lights on these unknown effects In particular, we demonstrate three main results: ) Incomplete information is chosen endogenously in the presence of exogenous uncertainty about the rival s behavior ) Incomplete information stabilizes the market which is unstable under full information

3 3) Incomplete information makes Parete-improvement by increasing the profits of both firms The remainder of the paper is organized as follows In Section, we present our mixed duopoly model In the first half of Section 3, we analyze the optimal behavior of the firms with full information in which the behavioral strategy of the firms is common knowledge Then, in the second half, we reexamine this model with partial information in which the behavioral strategy of the rival firm is unknown In Section 4, we compare the two types of duopoly equilibria with full information with partial information In Section 5, we construct discrete dynamic models in the fully-informed in the partially informed cases then show the main conclusions mentioned just above Section 6 concludes the paper Cournot-Bertr Competitions Mixed Duopoly Model There are two firms in a market in which firm i produces good x i with a constant unit production cost c i for i =, The inverse dem function of firm i is P i = α i β i x i γ i x j, () where α i > 0 β i > 0 By introducing notation p i = P i β i,a i = α i β i θ i = γ i β i, we obtain the simplified inverse dem functions, p i = A i x i θ i x j () Here A i denotes the maximum price to be attained when dem is zero is assumed to satisfy the followin conditions : Assumption (a) A = A = A; (b) A>max(c,c ) Assumption (a) is imposed on only for analytical simplicity Assumption (b) is imposed to guarantee interior optimum in the profit maximization of the firms The parameter θ i in () denotes the degree of product differentiation It is the ratio of γ i over β i where β i indicates the direct effect on price p i caused It is shown in Singh Vives (984) that this linear structure of inverse dem is obtained by maximizing the quadratic strictly concave utility function U(x,x )=α x + α x (β x +γ i x x + β x ) P i x i i= 3

4 by a change in x i γ i shows the indirect effect caused by a change in x j If θ i =, then both effects are the same hence the inverse dem functions for i =, become identical, which imply dems for homogeneous goods If θ i =0, then there is no indirect effect Thus the two goods are independent the duopoly market is divided into two monopoly markets A positive or negative θ i implies that the two goods are substitutes or complements In this study we confine our analysis to the case in which goods are substitutes the direct effect dominates the indirect effect To this end, we make the following assumption: Assumption 0 < θ i < Solving () with i =, for x x yields the direct dem of firm i, x i = {( θ i )A p i + θ i p j } (3) θ θ Each firm is either quantity-adjusting or price-adjusting maximizes its profit by taking its rival s behavior as given In particular, firm i maximizes its profit π i =(p i c i )x i (4) subject to either () if it is quantity adjusting or (3) if price-adjusting With two firms two strategies, there are four possible types of duopoly competition according to the strategy selection of the firms In this study, we focus on heterogeneous competition in which different firms take different strategies: in Cournot-Bertr (CB) competition, firm is quantity-adjusting, firm is price-adjusting in Bert-Cournot (BC ) competition, firm is priceadjusting firm is quantity-adjusting 3 Since Bertr-Cournot competition is dual to Cournot-Bertr competition, we restrict our study only to the optimal behavior of firms in CB competition As a benchmark, we start with the case with full information then proceed to the case of partial information The optimal values of outputs, prices profits of the firms under full information are obtained in Section those under partial information will be derived in Section 3 Full Information Case Under full information, firm knows that its rival is price-adjusting, sets its quantity x, taking its rival s strategic variable p as given Setting i = substituting () into (4) then substituting x from the solution of () give the profit offirm in terms of x p : π =(( θ )A ( θ θ )x + θ p c )x (5) Changing the sign of θ i from positive to negative, we can go from substitutes to complements Yousefi Szidarovszky(005) exhibits emergence of complex dynamics in a nonlinear differentiated duopoly when the indirect effect dominates the direct effect 3 The remaining two types are Cournot-Cournot competition in which the two firms are quantity-adjusting Bertrant-Bertr competition in which the two firms are price adjusting 4

5 Firm also knows that firm is quantity-adjusting, charges its price p by taking its rival s strategic variable x as given Solving () with i =for x then substituting it into (4) give the profit offirm in terms of x p : π =(p c )(A θ x p ) (6) Assuming interior optimum solving the first-order conditions for profit maximization of π i yields the following forms of the best reply functions: R CB θ (p )= ( θ θ ) p + ( θ )A c ( θ θ ) (7) R CB (x )= θ x + A + c (8) The superscript "CB" attached to functions variables indicates that they are given in a CB competition An intersection of these best reply functions is a solution of the simultaneous equations, x = R CB (p ) p = R CB (x ) In the (p,x ) plane, the x = R CB (p ) curve is positive-sloping the p = R CB (x ) curve is negativesloping Since both curves are linear, there is a unique solution We call the solution a CB equilibrium The outputs at the equilibrium are x CB = (A c ) θ (A c ) 4 3θ θ (9) x CB = ( θ θ )(A c ) θ (A c ) (0) 4 3θ θ The corresponding equilibrium prices are p CB = ( θ θ )(A c ) θ ( θ θ )(A c )+(4 3θ θ )A 4 3θ θ () p CB = θ (A c ) ( θ θ )(A c )+(4 3θ θ )A () 4 3θ θ Note that both are positive due to Assumption 4 Substituting the CB outputs prices into the profit functionspresentsthecb equilibrium profits: π CB =( θ θ ) x CB (3) 4 The numerators of () () are rewritten, respectively, as c ( θ θ )+c θ ( θ θ )+A( θ )( θ θ ) > 0 c θ +c ( θ θ )+A( θ ( + θ )) > 0 5

6 π CB = x CB, (4) showing that the profits are positive at the CB equilibrium Before proceeding further, we introduce an alternative expression of the cost ratio, c = A c, A c which is positive by Assumption (b) Solving this definition for c gives c = cc +( c)a which implies that c = c if c =,c >c if c< c <c if c> Clearly c is closely related to the cost ratio From (9) (0), the nonnegativity conditions for CB outputs are given as θ c (5) θ θ θ If the marginal cost, c, of firm is large enough to make the cost ratio exceed the threshold value /θ, then firm chooses zero-production or exits the market to avoid negative profit By the same token, if the marginal cost, c, of firm is large enough to make the cost ratio smaller than the threshold value θ /( θ θ ), then firm chooses zero-production or exists the market In either case, duopoly competition turns into monopoly Although it seems to be interesting to consider the birth of monopoly through the duopoly competition with the extreme cost ratio, we confine our analysis to the case when both firms stay in the market 3 Partial Information Case In CB competition with partial information (CB p competition henceforth), it is assumed that none of the firms have enough information to observe its rival s strategic behavior therefore each firm determines its best choice, presuming that its rival takes the same strategy That is, firm thinks that it is in a Cournot competition firm believes that it is in a Bertr competition The profit functions of the firms are therefore defined as follows Substituting () for i =into (4) gives the profit offirm in terms of its strategic variable x the believed strategic variable x of its competitor: π =(A x θ x c )x Differentiating π with respect to x gives the first-order condition for the profit maximization, which is then solved for x to obtain the best response of firm : x = A c θ x (6) 6

7 Firm thinks that it is in a Bertr competition so its strategic variable is p believes that its rival sets a price, p Substituting (3) for i =into (4) gives its profit: π =(p c ) ( θ )A + θ p p θ θ Differentiating π with respect to p gives the first-order condition, which is solved for p to derive the best response of firm : p = ( θ )A + c + θ p (7) Due to the partial information, each of expressions (6) (7) depends on the variable that the rival firm is supposed to choose To express both best response functions in terms of the decision variables, x p, of the firms, we solve () with i =for x (3) with i =for p to obtain x = A θ x p p =( θ )A ( θ θ )x + θ p Substituting them into (6) (7) provides the best response functions in terms of the decision variables, x p : R CB p (x,p ) ( θ )A c + θ θ x + θ p (8) R CB p (x,p ) ( θ θ )A + c θ ( θ θ )x + θ θ p (9) The superscript "CB p " is attached to functions variables to indicate that they are given in CB p competition The equilibrium point is defined by a pair of (x,p ) such that x = R CB p (x,p ) p = R CBp (x,p ) Solving these equations simultaneously using the dem function expressions yield the equilibrium outputs x CBp = ( θ θ )(A c ) θ (A c ) (0) 4 3θ θ x CB p = (A c ) θ (A c ), () 4 3θ θ where the nonnegativity conditions for the outputs are given by θ c θ θ () θ Using () with i = i =, we have the equilibrium prices, p CBp = ( θ θ )(A c ) θ (A c )+(4 3θ θ )A (3) 4 3θ θ 7

8 p CBp = θ ( θ θ )(A c ) ( θ θ )(A c )+(4 3θ θ )A 4 3θ θ, (4) both of which are positive The equilibrium profits are given as ³ π CB p = x CB p (5) ³ π CB p =( θ θ ) x CB p Notice that both profits are positive at the equilibrium 3 Information Difference (6) In this section, we compare the equilibrium strategies under full information with those under partial information to show the effects caused by information asymmetry We start with firm The output difference is obtained by subtracting (0) from (9), x CB x CB p = (A c )θ θ > 0 4 3θ θ the price difference by subtracting () from (3), p CB p CB p = (A c )θ θ 4 3θ θ (cθ ), where cθ may be positive, negative or even zero 5 The profit difference is given by subtracting (3) from (5), ³p ³p π CB π CB p = θ θ x CB + x CB p θ θ x CB x CB p, in which the first factor is positive the second factor can be rewritten as (A c ) θ θ ( θ θ ( cθ )) 4 3θ θ Since A c, θ θ the denominator are positive, h i sign π CB π CB p = sign[ p θ θ ( cθ )] (7) We will next show that π CB > π CB p under economically meaningful conditions 5 If c (ie, firm has a more efficient cost function), then p CB <p CBp 8

9 To determine the sign of the right h side of (7), we first suppose c Then p θ θ ( cθ ) p θ θ ( θ ) > 0, where the last inequality is due to relation θ θ > θ > θ for 0 < θ i < Therefore we have π CB > π CBp for c Next suppose that c< The π CB = π CB p locus is negative-sloping in θ defined by θ = c( cθ ), which implies that π CB > π CBp if θ <c( cθ ) π CB < π CBp if θ > c( cθ ) Returning to the nonnegativity condition of x CB solving it for θ yields that the x CB =0locus is θ = c +cθ Subtracting this equation from the equal-profit locus yields c( cθ ) c +cθ = c θ ( cθ ) +cθ > 0, since 0 < θ < 0 <c< This shows that the π CB = π CB p locus is locatedintheregioninwhichx CB < 0 Alternatively, π CB > π CBp in the feasible region in which x CB 0 for c< Combining this observation with thelastresultimpliesthatπ CB > π CB p inthefeasibleregionofθ in which x CB > 0 We now compare the equilibrium strategies of firm in CB competitions with both full partial informations As in the case of firm, the output difference price difference are obtained as follows: x CB x CB p = (A c )θ θ 4 3θ θ < 0 p CB p CB p = (A c )θ θ (c θ ), 4 3θ θ where c θ may be positive, negative or zero 6 The profit difference is given by ³ π CB π CB p = x CB + p ³ θ θ x CB p x CB p θ θ x CB p, where the first factor is positive the second factor can be rewritten as (A c ) θ θ (c( θ θ ) θ ) 4 3θ θ 6 If c> (ie, firm has a more efficient cost function), then p CB >p CB p 9

10 Since A c, θ θ the denominator are positive, h i sign π CB π CB p = sign[c( p θ θ ) θ ] (8) We can also show that π CB < π CB p under economically meaningful conditions To determine the sign of the right h side of (8), we first suppose that c Then c( p θ θ ) θ θ p θ θ < 0, since θ θ > θ > θ for 0 < θ i < Therefore, π CB < π CB p as c Next suppose that c> The π CB = π CBp locus is negative-sloping in θ defined by θ = c( cθ ), which implies that π CB < π CB p if θ <c( cθ ), π CB > π CB p if θ > c( cθ ) Returning to the nonnegativity condition of x CB p solving it for θ implies that the x CBp =0locus is θ = cθ θ Subtracting it from the equal profit locus yields c( cθ ) cθ = ( cθ )(cθ ) > 0 θ +cθ The last inequality is due to cθ > 0 which is shown as follows The x CBp =0locus crosses the horizontal line θ =at θ = +c, which implies that the feasible domain of the x CBp =0locusisaninterval[θ, ) For θ > θ, < c +c <cθ The direction of the inequality implies that the π CB = π CBp locus is located in theregioninwhichx CB p < 0 Alternatively, π CB < π CB p inthefeasibleregion in which x CB 0 for c> Combining this observation with the last result implies that π CB < πcb p inthefeasibleregionofθ θ, in which x CB p > 0 We can summarize the above result as follows: Theorem In the feasible parameter region in which x CB p > 0 x CB > 0, the quantity-adjusting firm earns more profit in the fully informed case than in the partially informed case (ie, π CB > π CBp ) while the price-adjusting firm earns more profit in the partially informed case than in the fully informed case(ie, π CB < π CB p ) 0

11 In comparing the equilibrium strategies under full information with those under partial information, an interesting phenomenon can be noticed: the equilibrium strategies under partial information are the duals of the equilibrium strategies under full information Comparing the equilibrium output levels (9) (0) with (0) (), the equilibrium prices () () with (3) (4), the equilibrium profits (3) (4) with (5) (6), we can see that they are identical if we interchange the two firms (ie, x CB i p CB i = p CBp j π CB i = x CBp j, = π CBp j for i, j =, i 6= j) Since the CB p ), competition is dual to the BC competition (π CBp = π BC π CBp = π BC Theorem can be alternatively summarized as π CB > π BC π CB < π BC, which is the result shown by Singh Vives (984) Although the equilibrium values are the same, it does not necessarily imply that the dynamic processes are also identical We will now turn our attention to the analysis of the dynamic processes 4 Stability Analysis In this section, we investigate the stability of the CB equilibrium that of the CB p equilibrium If the firms have naive expectations the time scale is discrete, then the best reply discrete dynamics under full information is constructed as follows: θ x (t +)= ( θ θ ) p (t)+ ( θ )A c ( θ θ ) (9) p (t +)= θ x (t)+ A + c This is a two dimensional linear system with Jacobi matrix θ 0 J CB ( θ θ ) = θ 0 The eigenvalues of the associated characteristic equation are λ, = ±i r θ θ, θ θ which are inside the unit circle if only if θ θ < 4 5 (30) In addition to this stability condition, we need to take the nonnegativity constraints (5) on the outputs, x CB x CB, into account, to obtain the proper condition for the stability of the CB equilibrium point:

12 Theorem In a CB competition with full information, the unique CB equilibrium is feasible stable if the degrees of the product differentiation, θ θ, the cost ratio, c, are in the following set: Θ CB = ½ (θ, θ,c) Θ θ θ < 4 5θ, θ θ θ c θ ¾ We now continue with examining the stability of the CB p competition Assuming naive expectations again, the best reply discrete dynamics under partial information is as follows: x (t +)= θ θ x (t)+ θ p (t)+ ( θ )A c (3) p (t +)= θ ( θ θ ) x (t)+ θ θ p (t)+ ( θ θ )A + c This dynamic system is also two-dimensional linear with Jacobi matrix θ θ θ J CB p = θ ( θ θ ) θ θ The characteristic equation is quadratic in λ, λ θ θ λ + θ θ =0 4 It is easy to verify that its eigenvalues are inside the unit circle since the stability conditions of this two dimensional system are satisfied under Assumption : detj CBp = θ θ > 0 4 ± trj CB p + detj CB p =± θ θ + θ θ > 0 4 In addition to these conditions, the outputs x CB p x CB p are subject to the nonnegativity conditions () Hence our second stability result can be summarized as follows: Theorem 3 In a CB competition with partial information, the unique CB p equilibrium is feasible stable if the degrees of the product differentiation, θ θ, c are in the following set: Θ CB p = ½ (θ, θ,c) θ c θ θ θ ¾

13 The parameter region in which equilibrium outputs are non-negative under CB CB p competitions is defined by ½ θ Θ = (θ, θ,c) c θ ¾ θ θ θ θ It can be divided into two parts by the θ θ =4/5 locus, with Θ s = ½ (θ, θ,c) Θ θ θ < 4 ¾ 5 Θ = Θ s Θ U Θ U = ½ (θ, θ,c) Θ θ θ 4 ¾ 5 As Figure shows, the lighter-gray region is Θ s, the darker-gray region is Θ U one or two non-negativity conditions are violated in the white regions Comparing these stability conditions with those under partial information, we first notice that less information improves stability since the fully-informed dynamic process is unstable while the partially-informed process is stable if (θ, θ,c) Θ U We then notice that collecting information on the rival s strategy by both firms to change the partially informed competition to fully informed competition is advantageous for firm since π CB > π CBp disadvantageous for firm since π CB < π CB p, regardless of whether the CB equilibrium is stable or unstable Figure Division of the feasible region by the stability condition As a result of profit maximizing behavior of the firms, it may be reasonable to assume that firm will collect information about the bahavioral strategy of its rival, make it known to firmaswellasfullyitsownbehavioralstrategy 3

14 to the competitor in order to change a partially informed oligopoly into the fully informed case However there is no guarantee that firm will believe trust firm in this message, so the price adjusting frim will continue believing that it is in a partially informed mixed oligopoly Hence CB competition with asymmetric information endogenously emerges In this casae, firm has the best reply function R CB (p ) firm has R CB p (x,p ) An intersection of these best reply functions is an equilibrium state under asymmetric informatition is denoted by (x CB a,p CB a ) with x CBa = ( θ θ )(A c ) θ (A c ) (4 θ θ )( θ θ ) (3) p CBa = A θ (A c )+(A c ) (33) 4 θ θ Similarly to (9) (3), the discrete dynamic system with naive expectations is given by x (t +)= θ ( θ θ ) p (t)+ ( θ )A c ( θ θ ) p (t +)= θ ( θ θ ) x (t)+ θ θ p (t)+ ( θ θ )A + c with Jacobi matrix J CB a = 0 θ ( θ θ ) θ ( θ θ ) θ θ The CB a equilibrium is asymptotically stable, since the stability conditions are satisfied: det J CB a > 0 ± trj CB a +detj CB a > 0 where det J CB a = θ θ trj CB a = θ θ 4 We summarize this result as follows: Theorem 4 In a CB competition with asymmetric information, the unique CB a equilibrium is feasible stable if (θ, θ,c) Θ This stability result implies that even if firm has full information firm has only partial information, they can still arrive at the CB a equilibrium Since this holds for (θ, θ,c) Θ U, we find again that less information stabilizes the otherwise unstable market under full information 4

15 The equilibrium values of the other variables are p CB a = (A c ) θ (A c )+(4 θ θ )A 4 θ θ (34) x CB a = θ (A c )+( θ θ )(A c ) (35) (4 θ θ )( θ θ ) We can also obtain the equilibrium profits of the two firms under asymmetric information: π CB a = (x CB a ) (36) θ θ π CBa = (x CBa ) (37) θ θ Then we have the following results concerning profit comparisons: Theorem 5 Equilibrium profits obtained at the CB a equilibrium are larger than the profits obtained at any other equilibria: π CBa > π CB π CBa > π CBp Proof Using (9), (0), (3) (35), it can be shown that x CB θ x CBa = θ (4 3θ θ )(4 θ θ )( θ θ ) xcb 0 x CB p x CB θ a = θ (4 3θ θ )(4 θ θ )( θ θ ) xcb p 0, where the inequalities are due to the non-negativity conditions of x CB x CB p Based on Assumption, the following profit ratios are greater than unity π CB a π CB = Ã ( θ θ ) x CB p x CB! > πcb a π CBp Ã! x CB a = > ( θ θ ) x CBp Since π CB > π CB p π CB p > π CB have been already shown in Theorem, this completes the proof 5 Concluding Remarks In this study we constructed mixed duopoly models with full information with partial information in which one firm is quantity-adjusting the other price-adjusting but each firm has uncertainty about the rival firm s behavioral strategy If the competition starts with partial information, firm finds it 5

16 profitable to collect information on the rival s strategic behavior (ie, quantityadjusting or price-adjusting), tells this knowledge to the competitor as well as informs it about its own strategic behavior to change the partially informed oligopoly into a fully informed competition If firm does not trust its rival in this message, it may continue believing that the oligopoly is still partially informed In consequence a CB competition with asymmetric information emerges Our main conclusions are summarized as follows: ) CB competition with asymmetric information is a natural consequence in the presence of exogenous uncertainty about the rival firm s behavior ) Asymmetric information stabilizes the market in the sense that the CB a equilibrium is stable even if (θ, θ,c) Θ U 3) Asymmetric information makes Pareto-improvement since both firms can make larger profits at the CB a equilibrium than at any other equilibria 6

17 References Bischi, G-I, A Naimzada L Sbragia, "Oligopoly Games with Local Monopolistic Approximation," Journal of Economic Behavior Organization, vol 6, , 007 Bischi, G-I, C Chiarella M Kopel, "The Long Run Outcomes Global Dynamics of a Duopoly Game with Misspecified Dem Functions," International Game Theory Review, vol 6, , 004 Bylka, S J Komar, "Cournot-Bertr Mixed Oligopolies," in M Beckman H P Künize eds, Warsaw Fall Seminars in Mathematical Economics 975 (Lecture Notes in Economics Mathematical Systems 33), Springer-Verlag, -33, 976 Dixit, A, "A Model of Duopoly Suggesting a Theory of Entry Barriers," Bell Journal of Economics, vol 0, 0-3, 979 Matsumoto, A, F Szidarovszky, "A Further Note on Price Quantity Competition in Differenitated Oligopolies," mimeo, 008 Matsumoto, A T Onozaki, "Heterogenous Strategies in Nonlinear Duopoly with Product Differentiation," Pure Mathematics Applications, vol 6, , 005 Singh, N X Vives, "Price Quantity Competition in a Differentiated Duopoly," R Journal of Economics, vol 5, , 984 Szidarovszky, F S Molnár, "Bertr, Cournot Mixed Oligopolies," Keio Economic Studies, vol 6, -8, 99 Yousefi, SFSzidarovszky,"OnceMoreonPriceQuantityCompetition in Differentiated Duopolies: A Simulation Study," Pure Mathematics Applications, vol 6, 55-53, 005 7

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